d/dx means “the derivative with respect to x”, an instruction to find how fast what follows changes as x changes. So d/dx [x³] = 3x², read as “the derivative of x cubed is three x squared”. The d stands for a tiny difference, which is why d/dx looks like a slope: a change in height divided by a change in x.

How do you differentiate a polynomial?

You differentiate a polynomial one term at a time, using the power rule on each term. For 3x² + 4x + 6, the 3x² becomes 6x, the 4x becomes 4, and the 6 disappears, because a constant never changes. So the derivative is 6x + 4, and its value at x = 2 is 6 × 2 + 4 = 16.

How do you remember the product rule?

Remember the product rule as taking turns: differentiate the first factor while the second waits, then the second while the first waits, and add the two results. In symbols, (fg)′ = f′g + fg′. Check it on x² · x³: it gives 2x · x³ + x² · 3x² = 5x⁴, exactly what you get by differentiating x⁵ directly.

Why do I keep getting the chain rule wrong?

Most chain rule errors come from forgetting the inside of the bracket. For (2x + 7)⁵, bringing down the 5 gives 5(2x + 7)⁴, but the inside, 2x + 7, also changes with x, at a rate of 2. Multiply by that 2 as well, and the derivative is 10(2x + 7)⁴. Leaving out the inner rate is the classic slip.

Can a calculus teacher use this derivative game for free?

A calculus teacher can, and it costs nothing: all six levels run in the browser of any school laptop or tablet, with no accounts or downloads. For a differentiation practice opener, draw y = x² and ask groups for the slope between x = 3 and x = 4, then 3.5, then 3.1. Their answers, 7, 6.5 and 6.1, close in on 6, the derivative at 3. With the schools plan, priced per student, a teacher dashboard shows who is improving.

Math practice · reimagined

Derivatives Game

This derivative game climbs from the power rule at x = 2 to partial derivatives, and every answer is a whole number from 2 to 200, never a formula. Differentiate faster than your rival, and your fighter hits while they are still bringing exponents down.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is a derivative in math?

A derivative is the rate at which a function changes at one exact point, and on a graph it is the slope of the curve there. We write it as f′(x) or d/dx. Think of a speedometer: as a car travels, its distance keeps growing, and the needle shows how fast it is growing at this very moment.

Every level of Derivative

Example: the derivative of 5x³ at x = 2

Bring the exponent 3 down in front and multiply it by the 5, which makes 15.
Then lower the exponent by one, from 3 to 2, so the derivative is 15x².
Now put in x = 2: 2² is 4, and 15 × 4 = 60, so you type 60.

Learn it step by step

A derivative measures how fast something is changing — the steepness of the curve at a point.

The power rule: d/dx of axⁿ is (a·n)·xⁿ⁻¹ — multiply by the exponent, then drop the exponent by one. So d/dx of 3x² is 6x; at x = 2 that's 12.

Apply the rule, plug in the point, and type the slope.

Tips and common mistakes

In the power rule, the power always drops by exactly one: x⁴ gives an x³ term, x² gives an x term, and a lone x gives a plain number.
A common slip is substituting too early. Put x = 2 into 4x³ first and you get 32, a constant whose derivative is 0. Differentiate first, then put in the point.
Sketch y = x² with a learner and hold a ruler against the curve at x = 1, 2 and 3. It tilts more each time, matching slopes 2, 4 and 6.

How to play

A typical round shows d/dx [4x³] at x=2, and you key in the derivative’s value there: 48. Chain and product rule levels want only the number in front: (3x + 1)⁵ differentiates to 15(3x + 1)⁴, so you type 15. Beat your rival with a correct answer and Broccoli attacks; the less time you took, the heavier the blow.

— Quick answers —

Derivative: the short FAQ

What does d/dx mean?: d/dx means “the derivative with respect to x”, an instruction to find how fast what follows changes as x changes. So d/dx [x³] = 3x², read as “the derivative of x cubed is three x squared”. The d stands for a tiny difference, which is why d/dx looks like a slope: a change in height divided by a change in x.
How do you differentiate a polynomial?: You differentiate a polynomial one term at a time, using the power rule on each term. For 3x² + 4x + 6, the 3x² becomes 6x, the 4x becomes 4, and the 6 disappears, because a constant never changes. So the derivative is 6x + 4, and its value at x = 2 is 6 × 2 + 4 = 16.
How do you remember the product rule?: Remember the product rule as taking turns: differentiate the first factor while the second waits, then the second while the first waits, and add the two results. In symbols, (fg)′ = f′g + fg′. Check it on x² · x³: it gives 2x · x³ + x² · 3x² = 5x⁴, exactly what you get by differentiating x⁵ directly.
Why do I keep getting the chain rule wrong?: Most chain rule errors come from forgetting the inside of the bracket. For (2x + 7)⁵, bringing down the 5 gives 5(2x + 7)⁴, but the inside, 2x + 7, also changes with x, at a rate of 2. Multiply by that 2 as well, and the derivative is 10(2x + 7)⁴. Leaving out the inner rate is the classic slip.
Can a calculus teacher use this derivative game for free?: A calculus teacher can, and it costs nothing: all six levels run in the browser of any school laptop or tablet, with no accounts or downloads. For a differentiation practice opener, draw y = x² and ask groups for the slope between x = 3 and x = 4, then 3.5, then 3.1. Their answers, 7, 6.5 and 6.1, close in on 6, the derivative at 3. With the schools plan, priced per student, a teacher dashboard shows who is improving.

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